Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-inftyexpiinv Structured version   Visualization version   GIF version

Theorem bj-inftyexpiinv 32096
Description: Utility theorem for the inverse of inftyexpi. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
Assertion
Ref Expression
bj-inftyexpiinv (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)

Proof of Theorem bj-inftyexpiinv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opeq1 4334 . . . 4 (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2 df-bj-inftyexpi 32095 . . . 4 inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3 opex 4853 . . . 4 𝐴, ℂ⟩ ∈ V
41, 2, 3fvmpt 6176 . . 3 (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
54fveq2d 6092 . 2 (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = (1st ‘⟨𝐴, ℂ⟩))
6 cnex 9874 . . 3 ℂ ∈ V
7 op1stg 7049 . . 3 ((𝐴 ∈ (-π(,]π) ∧ ℂ ∈ V) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴)
86, 7mpan2 702 . 2 (𝐴 ∈ (-π(,]π) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴)
95, 8eqtrd 2643 1 (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  cop 4130  cfv 5790  (class class class)co 6527  1st c1st 7035  cc 9791  -cneg 10119  (,]cioc 12006  πcpi 14585  inftyexpi cinftyexpi 32094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fv 5798  df-1st 7037  df-bj-inftyexpi 32095
This theorem is referenced by:  bj-inftyexpiinj  32097
  Copyright terms: Public domain W3C validator